If two events A and B are such that P,(A + B) = frac{5}{6}, P,(AB) = frac{1}{3}, and P,(bar A) = fra

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14.Probability

easy

If two events $A$ and $B$ are such that $P\,(A + B) = \frac{5}{6},$ $P\,(AB) = \frac{1}{3}\,$ and $P\,(\bar A) = \frac{1}{2},$ then the events $A$ and $B$ are

A

Independent

B

Mutually exclusive

C

Mutually exclusive and independent

D

None of these

Solution

(a) We have $P(A + B) = P(A) + P(B) – P(AB)$

$ \Rightarrow \frac{5}{6} = \frac{1}{2} + P(B) – \frac{1}{3} $

$\Rightarrow P(B) = \frac{4}{6} = \frac{2}{3}$

Thus, $P(A)\,.\,P(B) = \frac{1}{2} \times \frac{2}{3} = \frac{1}{3} = P(AB)$

Hence events $A$ and $B$ are independent.

Standard 11

Mathematics

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